The number of points of local maxima of the function $f(x) = x + \sin x$ is-

Let $f(x) = \frac{\sin \pi x}{x^2}, x > 0$. Let $x_1 < x_2 < x_3 < \ldots < x_n < \ldots$ be all the points of local maximum of $f(x)$ and $y_1 < y_2 < y_3 < \ldots < y_n < \ldots$ be all the points of local minimum of $f(x)$. Which of the following statements are true?
$(1)$ $|x_n - y_n| > 1$ for every $n$
$(2)$ $x_1 < y_1$
$(3)$ $x_n \in (2n, 2n + \frac{1}{2})$ for every $n$
$(4)$ $x_{n+1} - x_n > 2$ for every $n$

If $x$ and $y$ are two positive real numbers such that $xy = 4$, then the minimum value of $\left(\sqrt{x} + \frac{y^2}{2}\right)$ is

Let $f(x)=1+\frac{x}{1 !}+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\frac{x^4}{4 !}$. The number of real roots of $f(x)=0$ is

$P(x) = x^4 + ax^3 + bx^2 + cx + d$ is such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

The function $f(x) = \sin x (1 + \cos x)$ is maximum at $x = \dots$